{ "metadata": { "name": "sympy_quantum_computing" }, "nbformat": 3, "worksheets": [ { "cells": [ { "cell_type": "markdown", "source": [ "# Basic Symbolic Quantum Mechanics with [SymPy](http://sympy.org)", "", "We first load the IPython extensions that enable LaTeX-based mathematical printing ", "of SymPy objects, and then import the quantum computing libraries from SymPy." ] }, { "cell_type": "code", "collapsed": true, "input": [ "%load_ext sympyprinting" ], "language": "python", "outputs": [], "prompt_number": 1 }, { "cell_type": "code", "collapsed": true, "input": [ "from sympy import sqrt, symbols, Rational", "from sympy import expand, Eq, Symbol, simplify, exp, sin", "from sympy.physics.quantum import *", "from sympy.physics.quantum.qubit import *", "from sympy.physics.quantum.gate import *", "from sympy.physics.quantum.grover import *", "from sympy.physics.quantum.qft import QFT, IQFT, Fourier", "from sympy.physics.quantum.circuitplot import circuit_plot" ], "language": "python", "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "source": [ "

Bras and Kets

" ] }, { "cell_type": "markdown", "source": [ "Create symbolic states" ] }, { "cell_type": "code", "collapsed": true, "input": [ "phi, psi = Ket('phi'), Ket('psi')", "alpha, beta = symbols('alpha beta', complex=True)" ], "language": "python", "outputs": [], "prompt_number": 3 }, { "cell_type": "markdown", "source": [ "Create a superposition" ] }, { "cell_type": "code", "collapsed": false, "input": [ "state = alpha*psi + beta*phi; state" ], "language": "python", "outputs": [ { "latex": [ "$$\\alpha {\\left|\\psi\\right\\rangle } + \\beta {\\left|\\phi\\right\\rangle }$$" ], "output_type": "pyout", "png": 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"prompt_number": 4, "text": [ "\u03b1\u22c5\u2758\u03c8\u27e9 + \u03b2\u22c5\u2758\u03c6\u27e9" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "source": [ "Dagger the superposition and multiply the original" ] }, { "cell_type": "code", "collapsed": false, "input": [ "ip = Dagger(state)*state; ip" ], "language": "python", "outputs": [ { "latex": [ "$$\\left(\\overline{\\alpha} {\\left\\langle \\psi\\right|} + \\overline{\\beta} {\\left\\langle \\phi\\right|}\\right) \\left(\\alpha {\\left|\\psi\\right\\rangle } + \\beta {\\left|\\phi\\right\\rangle }\\right)$$" ], "output_type": "pyout", "png": 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"prompt_number": 5, "text": [ "", "\u239b_ _ \u239e ", "\u239d\u03b1\u22c5\u27e8\u03c8\u2758 + \u03b2\u22c5\u27e8\u03c6\u2758\u23a0\u22c5(\u03b1\u22c5\u2758\u03c8\u27e9 + \u03b2\u22c5\u2758\u03c6\u27e9)" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "source": [ "Distribute" ] }, { "cell_type": "code", "collapsed": false, "input": [ "qapply(expand(ip))" ], "language": "python", "outputs": [ { "latex": [ "$$\\alpha \\overline{\\alpha} \\left\\langle \\psi \\right. {\\left|\\psi\\right\\rangle } + \\alpha \\overline{\\beta} \\left\\langle \\phi \\right. {\\left|\\psi\\right\\rangle } + \\beta \\overline{\\alpha} \\left\\langle \\psi \\right. {\\left|\\phi\\right\\rangle } + \\beta \\overline{\\beta} \\left\\langle \\phi \\right. {\\left|\\phi\\right\\rangle }$$" ], "output_type": "pyout", "png": 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"prompt_number": 6, "text": [ "", " _ _ _ _ ", "\u03b1\u22c5\u03b1\u22c5\u27e8\u03c8\u2758\u03c8\u27e9 + \u03b1\u22c5\u03b2\u22c5\u27e8\u03c6\u2758\u03c8\u27e9 + \u03b2\u22c5\u03b1\u22c5\u27e8\u03c8\u2758\u03c6\u27e9 + \u03b2\u22c5\u03b2\u22c5\u27e8\u03c6\u2758\u03c6\u27e9" ] } ], "prompt_number": 6 }, { "cell_type": "markdown", "source": [ "

Operators

" ] }, { "cell_type": "markdown", "source": [ "Create symbolic operators" ] }, { "cell_type": "code", "collapsed": true, "input": [ "A = Operator('A')", "B = Operator('B')", "C = Operator('C')" ], "language": "python", "outputs": [], "prompt_number": 7 }, { "cell_type": "markdown", "source": [ "Test commutativity" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A*B == B*A" ], "language": "python", "outputs": [ { "output_type": "pyout", "prompt_number": 8, "text": [ "False" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "source": [ "Distribute A+B squared" ] }, { "cell_type": "code", "collapsed": false, "input": [ "expand((A+B)**2)" ], "language": "python", "outputs": [ { "latex": [ "$$A B + \\left(A\\right)^{2} + B A + \\left(B\\right)^{2}$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAMAAAAAZCAYAAABn7SHgAAAABHNCSVQICAgIfAhkiAAABHpJREFU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\n/wF5Fp03jGX3sQAAAABJRU5ErkJggg==\n", "prompt_number": 9, "text": [ "", " 2 2", "A\u22c5B + A + B\u22c5A + B " ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "source": [ "Create a commutator" ] }, { "cell_type": "code", "collapsed": false, "input": [ "comm = Commutator(A,B); comm" ], "language": "python", "outputs": [ { "latex": [ "$$\\left[A,B\\right]$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAAWCAYAAAC/kK73AAAABHNCSVQICAgIfAhkiAAAAj5JREFU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\niuR3f19QpToAAAAASUVORK5CYII=\n", "prompt_number": 10, "text": [ "[A,B]" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "source": [ "Carry out the commutator" ] }, { "cell_type": "code", "collapsed": false, "input": [ "comm.doit()" ], "language": "python", "outputs": [ { "latex": [ "$$A B - B A$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAASCAYAAAD15uiRAAAABHNCSVQICAgIfAhkiAAAAi9JREFU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\nRK5CYII=\n", "prompt_number": 11, "text": [ "A\u22c5B - B\u22c5A" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "source": [ "Create a more fancy commutator" ] }, { "cell_type": "code", "collapsed": false, "input": [ "comm = Commutator(A*B,B+C); comm" ], "language": "python", "outputs": [ { "latex": [ "$$\\left[A B,B + C\\right]$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAWCAYAAAA/45nkAAAABHNCSVQICAgIfAhkiAAAA5VJREFU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\nASen61ooc9yoAAAAAElFTkSuQmCC\n", "prompt_number": 12, "text": [ "[A\u22c5B,B + C]" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "source": [ "Expand the commutator" ] }, { "cell_type": "code", "collapsed": false, "input": [ "comm.expand(commutator=True)" ], "language": "python", "outputs": [ { "latex": [ "$$\\left[A,B\\right] B + \\left[A,C\\right] B + A \\left[B,C\\right]$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAN8AAAAWCAYAAABAHklQAAAABHNCSVQICAgIfAhkiAAABNdJREFU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\ngg==\n", "prompt_number": 13, "text": [ "[A,B]\u22c5B + [A,C]\u22c5B + A\u22c5[B,C]" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "source": [ "Carry out and expand the commutators" ] }, { "cell_type": "code", "collapsed": false, "input": [ "_.doit().expand()" ], "language": "python", "outputs": [ { "latex": [ "$$A B C + A \\left(B\\right)^{2} - B A B - C A B$$" ], "output_type": "pyout", "png": 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"prompt_number": 14, "text": [ "", " 2 ", "A\u22c5B\u22c5C + A\u22c5B - B\u22c5A\u22c5B - C\u22c5A\u22c5B" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "source": [ "Take the dagger" ] }, { "cell_type": "code", "collapsed": false, "input": [ "Dagger(_)" ], "language": "python", "outputs": [ { "latex": [ "$$- B^{\\dagger} A^{\\dagger} B^{\\dagger} - B^{\\dagger} A^{\\dagger} C^{\\dagger} + \\left(B^{\\dagger}\\right)^{2} A^{\\dagger} + C^{\\dagger} B^{\\dagger} A^{\\dagger}$$" ], "output_type": "pyout", "png": 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"prompt_number": 15, "text": [ "", " 2 ", " \u2020 \u2020 \u2020 \u2020 \u2020 \u2020 \u239b \u2020\u239e \u2020 \u2020 \u2020 \u2020", "- B \u22c5A \u22c5B - B \u22c5A \u22c5C + \u239dB \u23a0 \u22c5A + C \u22c5B \u22c5A " ] } ], "prompt_number": 15 } ] } ] }